This problem involves the concept of work and time, specifically an inverse proportionality relationship between the number of men and the time taken to complete a fixed amount of work. We can use the formula \(M_1 D_1 = M_2 D_2\), where \(M\) represents the number of men and \(D\) represents the number of days.
Therefore, 264 men are required to complete the same work in 8 days.
Correct Option: B) 264
The question asks for the most typical industrial method to produce sodium hydroxide (NaOH). We need to identify the process specifically designed for NaOH production and distinguish it from processes used for other industrial chemicals.
Sodium hydroxide, also known as caustic soda, is a strong base widely used in various industries. Its industrial production primarily involves the electrolysis of an aqueous solution of sodium chloride (brine).
During the electrolysis of brine, the following reactions occur:
This process is known as the chlor-alkali process, and it is the dominant method for industrial production of sodium hydroxide, chlorine, and hydrogen.
C) Electrolysis of brine — This is the primary industrial method for producing sodium hydroxide (NaOH), along with chlorine gas (\(\text{Cl}_2\)) and hydrogen gas (\(\text{H}_2\)).
The pH scale measures the acidity or alkalinity of a solution. Pure water is considered neutral, meaning it is neither acidic nor basic. The pH of pure water is determined by the autoionization of water and the equilibrium constant for this process.
Autoionization of Water: Water molecules can react with each other in a process called autoionization (or autoprotolysis) to form hydronium ions (\(\text{H}_3\text{O}^+\) or simply \(\text{H}^+\)) and hydroxide ions (\(\text{OH}^-\)).
\[ \text{H}_2\text{O} \text{(l)} + \text{H}_2\text{O} \text{(l)} ightleftharpoons \text{H}_3\text{O}^+ \text{(aq)} + \text{OH}^- \text{(aq)} \]Ion Product of Water (\(K_w\)): At 25°C, the equilibrium constant for this reaction, known as the ion product of water (\(K_w\)), is approximately \(1.0 \times 10^{-14}\).
\[ K_w = [\text{H}_3\text{O}^+][\text{OH}^-] = 1.0 \times 10^{-14} \text{ at } 25^\circ\text{C} \]Concentrations in Pure Water: In pure water, the concentration of hydronium ions is equal to the concentration of hydroxide ions because they are produced in a 1:1 ratio.
\[ [\text{H}_3\text{O}^+] = [\text{OH}^-] \]Calculate \([\text{H}_3\text{O}^+]\): Substitute this into the \(K_w\) expression:
\[ [\text{H}_3\text{O}^+]^2 = 1.0 \times 10^{-14} \]\[ [\text{H}_3\text{O}^+] = \sqrt{1.0 \times 10^{-14}} = 1.0 \times 10^{-7} \text{ M} \]Calculate pH: The pH is defined as the negative base-10 logarithm of the hydronium ion concentration.
\[ \text{pH} = -\log_{10}[\text{H}_3\text{O}^+] \]\[ \text{pH} = -\log_{10}(1.0 \times 10^{-7}) \]\[ \text{pH} = -(-7) = 7 \]Therefore, the pH value of pure water at 25°C is 7, which indicates a neutral solution.
B) 7 — Pure water is neutral, meaning it has an equal concentration of hydrogen ions (\(\text{H}^+\)) and hydroxide ions (\(\text{OH}^-\)). At 25°C, this concentration is \(1.0 \times 10^{-7}\) M for both, leading to a pH of 7.
A) 3 — A pH of 3 indicates an acidic solution. Solutions with pH values less than 7 are acidic.
C) 5 — A pH of 5 also indicates an acidic solution, though less acidic than a solution with pH 3.
D) 1 — A pH of 1 indicates a strongly acidic solution. Solutions with pH values much lower than 7 are highly acidic.
This problem involves the Tangent-Secant Theorem, which states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external segment. We will use this theorem to find the length of the secant segment PN and then calculate the length of the chord MN.
D) 18 cm
This is a syllogism problem. We need to analyze the given statements and determine which conclusions logically follow from them. We can use Venn diagrams or logical deductions to solve this.
Correct Option: D) Only conclusion (I) follows.
The question asks for the location where the Indian Air Force (IAF) successfully tested its Arogya Maitri Disaster Management Cube in May 2024. This is a factual question related to recent defense and disaster management developments in India.
Correct Option: B) Agra
This problem requires us to substitute the given interchange of mathematical operators into the expression and then evaluate it using the BODMAS/PEMDAS rule (Order of Operations).
B) 26 is the correct answer after applying the operator interchanges and following the order of operations.
The average price of three items of furniture is Rs 16,845. If their prices are in the ratio 3:5:7, the price of the cheapest item (in Rs) is:
To find the price of the cheapest item, we first need to determine the total price of the three items using their average price. Then, we can use the given ratio to find the individual prices and identify the cheapest one.
Calculate the total price of the three items:
The average price of three items is Rs 16,845. The total price is the average price multiplied by the number of items.
Represent the prices using the given ratio:
The prices of the three items are in the ratio 3:5:7. Let the prices be \(3x\), \(5x\), and \(7x\) respectively.
Form an equation and solve for \(x\):
The sum of the individual prices must equal the total price.
Calculate the price of the cheapest item:
The cheapest item corresponds to the smallest ratio part, which is \(3x\).
Calculate the prices of the other items (optional, for verification):
Second item price: \(5x = 5 \times 3,369 = \text{Rs } 16,845\)
Third item price: \(7x = 7 \times 3,369 = \text{Rs } 23,583\)
Check total: \(10,107 + 16,845 + 23,583 = \text{Rs } 50,535\). This matches the total price calculated in step 1.
Correct Option: D) 10,107
The question asks to identify which category is NOT a class of consumers. To answer this, we need to understand the definitions of producers and consumers in an ecosystem.
D) Producers — Producers are autotrophs; they synthesize their own food, typically through photosynthesis, and do not consume other organisms. Therefore, they are not a class of consumers.